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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the $ X$-rank of a curve $ X\subset\mathbb{P}^n$: an extremal case


Author: E. Ballico
Journal: Proc. Amer. Math. Soc. 141 (2013), 1211-1213
MSC (2010): Primary 14H99, 14N05
Published electronically: August 29, 2012
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Abstract: Let $ X\subset \mathbb{P}^n$, $ n \ge 3$, be an integral and non-degenerate curve. For any $ P\in \mathbb{P}^n$ the $ X$-rank $ r_X(P)$ of $ P$ is the minimal cardinality of a set $ S\subset Y$ such that $ P$ is in the linear span of $ S$. Landsberg and Teitler proved that $ r_X(P) \le n$ for any $ X$ and any $ P$. Here we classify the pairs $ (X,Q)$, $ Q\in X_{reg}$, such that all points of the tangent line $ T_QX$ (except $ Q$) have
$ X$-rank $ n$: $ X \cong \mathbb{P}^1$ and $ T_QX$ has order of contact $ \deg (X) +2-n$ with $ X$ at $ Q$.


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Additional Information

E. Ballico
Affiliation: Department of Mathematics, University of Trento, 38123 Povo (TN), Italy
Email: ballico@science.unitn.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11406-6
PII: S 0002-9939(2012)11406-6
Keywords: Ranks, projective curve, tangent developable, symmetric tensor rank, $X$-rank
Received by editor(s): June 25, 2011
Received by editor(s) in revised form: August 18, 2011
Published electronically: August 29, 2012
Additional Notes: The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
Communicated by: Irena Peeva
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.